Spatial referencing

An overview

Richard Knippers

Learning instructions

Learning activities:

� Literature: ITC Core textbook, Chapter 3.1 on Spatial Referencing.

� Website: http://kartoweb.itc.nl/geometrics

� Exercise: Spatial referencing (ArcMap10)

Questions: Blackboard Discussion Board

Main objectives

� Understand the relevance and actual use of reference surfaces, coordinate systems, and coordinate transformations in mapping.

� Describe and differentiate between coordinate systems and map projections.

� Grasp the logic of map projection equations and the principles of transforming maps from one projection system to another.

Contents

� Spatial reference surfaces and datums

� The Geoid – vertical (height) datum

� The Ellipsoid – horizontal (geodetic) datum

� Local and global datums

� Map projections

� Classification of map projections

� Map projection selection

� Map coordinate systems (e.g. UTM)

� Coordinate transformations

Cone

Plane

Ellipsoid (best fitting)

Geodetic (or horizontal) datum

The Earth: a complex shape

Mapping surface

Independent handling of

horizontal and vertical

Cylinder

Earth to Map

The ellipsoid

The Earth

Independent handling of

horizontal and vertical

The Geoid

Reference surfaces for mapping

Spatial reference surface -

The Geoid

The Geoid – Vertical (height) datum

The Earth

The Geoid

Global Sea Level

Starting from Mean Sea Level (MSL) points, the heights (H) of points on the Earth can be measured using a technique

known as geodetic leveling.

Geodetic levelling

H = Orthometric height

H

Tide-gauge

benchmark

(zero height)

Vertical datums

Every country (or group of countries) has it’s own

Mean Sea Level - its own vertical (height) datum.

The Netherlands

Belgium

MSL of Belgium is 2.34m lower than MSL of The Netherlands

At what height do we live?http://www.ahn.nl/postcodetool

Elevation data are related to Amsterdam Zero (N.A.P.)

HNAP = 37m

GPS height versus N.A.P. height

hWGS84 = 107.5m

HNAP = hWGS84 - 44m (N) - 27m (ITC building) = 37m

Ellipsoidal heightOrthometric height

(height above the Geoid)

Ellipsoidal height versus Orthometric height

Relation between Geoid and Ellipsoid

The earth’s surface, and the geoid and a reference ellipsoid used to

approximate it. The geoidal undulation (N) is the separation between

the geoid and an ellipsoid. It varies globally between ±110 m.

N

H h

N

H = Orthometric height

h = Ellipsoidal height

N = Geoidal separation (undulation)

Trends in mapping: global vertical datums

Satellite gravity missions (e.g. GOCE) make it possible to determine a global vertical datum with centimetres

accuracy.

Spatial reference surface -

The Ellipsoid

The Ellipsoid

Typical values of the parameters for an ellipsoid:

a = 6378137.0 m b = 6356752.31 m

f = 1/298.26 e = 0.0818187

Flattening: Eccentricity:

f = (a-b)/a e2 = (a2 - b2)/a2

18

Geographic coordinates (ϕ, λ)

φ = 52°9' 18.62"

λ = 5°23' 13.93"φ = 52°9' 18.62"

λ = 5°23' 13.93"h = 67m

φ (phi) = latitude angle

λ (lambda) = longitude angle

Local and global ellipsoids/datums

The ellipsoid is chosen in such way that it best fits the surface of the area of interest (the

country)

e.g. WGS84

e.g. Amersfoort

(Bessel)

Local datum system of the Netherlands

Ellipsoid

Bessel ellipsoid

Amersfoort datum

National triangulation network

φA= 52°13' 26.2"N

λA= 06°53' 32.1"E

XRD = 257776.47m

YRD = 471588.14m

Geodetic datum

φA= 52°9' 22.2“

λA= 5°23' 15.5"

XRD = 155 000m

YRD = 463 000m

Local datum systems

Countries (or regions) use there own datum system to make accurate maps.

Local and global ellipsoids/datums

The ellipsoid is chosen in such way that it best fits the surface of the area of interest (the

country)

e.g. WGS84

e.g. Amersfoort

(Bessel)

International Terrestrial Reference System (ITRS)

International Terrestrial Reference Frame (ITRF)

International Terrestrial Reference System (ITRS)

Global reference system for global surveying and mapping. The system uses the GRS80 (or WGS84) ellipsoid

x = 3903583m

y = 368117m

z = 5012955m

xITRF2000= 5125124.34m

yITRF2000= 1998556.28m

zITRF2000= -3217389.44m

φITRF2000= 30°29'24.62"SλITRF2000= 21°18'12.36"E

hITRF2000 =145.25m

φITRF2008= 30°29'24.61"SλITRF2008= 21°18'12.35"E

hITRF2008 =145.17m

xITRF2008= 5125124.42m

yITRF2008= 1998556.15m

zITRF2008= -3217389.23m

Trends in mapping: global horizontal datums

Global ellipsoids and datumsto approximate the earth-as-a-whole - with the

aid of satellites- are becoming more in use

(e.g. WGS84, ITRF, ETRS89).

Changing or re-adjustment of local ellipsoids and datums is taking place in many countries.

Map projections

26

Map projection principle

2D Cartesian coordinates (X,Y)

Geographic coordinates (φ,λ)

longitude angle

latitude

angle

φ = 55°00' 00" N

λ = 45°00' 00" W

x = 280,000m

y = 310,000m

Map projection equations

Forward equation

A map projection is a mathematical function by which 2D

Geographic coordinates(φ,λ) are transformed into 2D Cartesian map coordinates (x,y)

Inverse equation

Map projection equations for the Mercator projection (spherical assumption)

Inverse mapping equation:

Forward mapping equation:

Map projection equations (example)

Mercator projectionConformal cylindrical projection

Scale distortions on the Mercator projection

Area distortions are significant towards the polar regions.

Greenland appears to be larger but is only one-eighth the size of

South America.

Scale distortions on a map

Areas smaller than 25 x 25 km: NO DISTORTIONS

Areas larger than 25x25 km: ALWAYS DISTORTIONS

Any map projection is associated with scale distortions. The amount

and kind of distortions depend on the type of map projection.

Map projection properties

ConformalAngles and shapes are correctly represented (locally)

Equivalent (or equal-area)Areas are correctly represented

EquidistantDistances from 1 or 2 points or along certain lines are

correctly represented

Cylindrical equal-area projection

Areas are correctly represented

Equidistant cylindrical projection(also called Plate Carrée)

Equidistant along the meridians

Robinson projectionPseudo-Cylindrical

Neither conformal nor equal-area (both shape and area are reasonably well preserved)

Map projection class

Map Projection surface

Ellipsoid

Cylindrical Projection Conical projection Azimuthal projection

Mercator Lambert Conformal Conical Stereographic projection

Classification of map projections

� Class

� Cylindrical

� Conical

� Azimuthal

� Secant or tangent

projection plane

� Property

� Equivalent (or equal-area)

� Equidistant

� Conformal

� Aspect (orientation)

� Normal

� Oblique

� Transverse

Secant projection planes

Classification of map projections

CYLINDRICAL

Oblique TransverseNormal

Tangent

Secant

Equidistant ConformalEquivalent

Conformal cylindrical projection with a transverse cylinder and secant projection plane (e.g. Universal Transverse Mercator)

Classification of map projections

CONICAL

Oblique TransverseNormal

Tangent

Secant

Equidistant ConformalEquivalent

Conformal conical projection with a normal cone and tangent projection plane (e.g. Lambert conformal conic)

Classification of map projections

AZIMUTHAL

Oblique TransverseNormal

Tangent

Secant

Equidistant ConformalEquivalent

Conformal azimuthal projection with a tangent polar projection

Plane (e.g. Universal Polar Stereographic)

Polar Equatorial

Selection of a Map projection (I)

Normal cylindrical projections are typically used to map theWorld in its entirety. Conical projections are often used tomap the different continents, while the normal azimuthal

projection may be used to map the polar areas.

Also consider the shape of the area to be mapped:

Selection of a Map projection (II)

ConformalMaps which require measuring angles (e.g. aeronautical

charts, topographic maps)

Equivalent (or equal-area)Maps which require measuring areas

(e.g distribution maps)

EquidistantMaps which require reasonable area and angle distortions

(e.g. several thematic maps)

Selection of a Map projection (III)

Oblique cylinder

The position (and orientation) of the projection plane is optimal

when the projection plane is located along the main axis of the area to be mapped, or when the projection centre coincides

with centre of the area.

Dutch RD system (RijksDriehoekstelsel)

Dutch map coordinate system

Projection: StereographicGeodetic datum: Amersfoort (Bessel ellipsoid)

Bessel ellipsoid

oblique secant mapping plane

Universal Transverse Mercator

International Standard

Universal Transverse Mercator projection

Conformal Cylindrical (transverse

secant) projection

UTM Mapping zones

80oS

84oN

Longitudinal

zone of 6o

UTM Mapping zones

0oEquator

Central Meridian

Gre

enw

ich

0o 6o

1………………………………… 29 30 31 32 ………………………60

Two adjacent UTM zones

UTM grid on maps

Topographic map of the Netherlands (scale 1:25,000)

GraticuleDutch RD grid

UTM grid

Coordinate transformations

Position City hall of Enschede

Position in Geographic coordinates:

� φφφφAmersfoort = 520 13' 26.2"N λλλλAmersfoort = 6

0 53' 32.1" E

(φ(φ(φ(φBessel = 52.2239440N λλλλBessel = 6.8922489

0E )

� φφφφETRS89 = 520 13' 22.6"N λλλλETRS89 = 60 53' 29.7" E

� φφφφWGS84 = 520 13' 22.6"N λλλλWGS84 = 60 53' 29.7" E

Position in Map (plane rectangular) coordinates:

� XDutch RD = 257790.12m YDutch RD = 471607.17m (Old RD1918)

� XDutch RD = 257776.47m YDutch RD = 471588.14m (New RD)

� XUTM31 = 765872.57m YUTM31 = 5793185.04m

� XUTM32 = 356065.01 m YUTM32 = 5788133.6m

Map projection changeUsing projection equations

Inverse equations Forward equations

Map projection changeIncluding a datum transformation

(e.g. ITRF) (e.g. Potsdam)

Inverse equations Forward equations

Datum shifts

B

Coordinate transformations (overview)

Projection unknown

Conformal, Affine or

Polynomial transformation

Projection change using a 2D Cartesian transformation

Control point

Projection B

The unknown coordinate system is related to a known

coordinate system on the basis of a set of known points

Application: Image Rectification (I)

True position

Measured position

error vector

Application: Image Rectification (II)

Georeferencing

Image Rectification (geocoding)

X

Y

X

Y

Application: Matching data layers

INTERNATIONAL INSTITUTE FOR GEO-INFORMATION SCIENCE AND EARTH OBSERVATION

Thank you!